# Study of the Geometry of an Oscillating Water Column Device with Five Chambers Coupled under Regular Waves through the Constructal Design Method

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

_{n}) and the length (L

_{n}) of the hydropneumatic chambers (H

_{n}/L

_{n}, where n varies from one to five). Based on the results of the mass flow rate and pressure, their influence on power was evaluated. It was observed that the influence of the degrees of freedom on the pressure difference, mass flow rate, and hydrodynamic power was quite similar, displaying an increase for low ratios of H

_{n}/L

_{n}up to a maximum magnitude and followed by a decrease in magnitude. The best performance was achieved for the geometric configuration with H

_{n}/L

_{n}= 0.2613 (H

_{n}= 5.0625 m and L

_{n}= 15.8219 m), representing an improvement of 98.6% compared to the worst case analyzed.

## 1. Introduction

#### State of the Art

_{n}/L

_{n}ratio (the ratio of the height to length of coupled hydropneumatic chambers). The results show that there is a maximum hydropneumatic power, with this value used as a reference for the analyzed configuration and geometric arrangement. This work, together with others, opened up space for the discussion of adding coupled chambers and device design and their influence on hydropneumatic power gain.

_{n}/L

_{n}(the ratio between the height and length of coupled hydropneumatic chambers), where n varies depending on the number of chambers, in this case, n = 1, …, 5. Due to its numerical nature, certain simplifications are taken into account, including a two-dimensional, transient, and incompressible multiphase flow.

## 2. Mathematical Modeling

^{−1}), and h is the depth (m).

#### 2.1. Description of Wave Characterization

_{n}), the submersion depth of the device (H

_{depth}), the elevations of the turbine ducts (H

_{j}), the lengths of the ducts (l

_{j}), the lengths of the chambers (L

_{n}), and the thicknesses of the columns dividing the devices (e

_{n}). Here, the variable n ranges from one to five, while j ranges from 7 to 11.

#### 2.2. Formulation of Pressure and Mass Flow Rate Using the RMS Methodology

_{water}is the amount of water in each volume, and A

_{t}is the area of each volume (m

^{2}). The monitoring of the mass flow rate at the center of each turbine duct is conducted through the following integral [29]:

_{air}is the density of the air (kg/m

^{3}). The mean values were calculated using the arithmetic mean for transient RMS problems [17]:

_{air}is the static pressure in the turbine duct of the OWC device (Pa), $\dot{m}$ is the mass flow rate through the turbine duct (kg/s), and v

_{air}is the velocity of the air in the turbine duct (m/s); the other parameters can be found in Gomes et al. [28] and Lima et al. [32].

#### 2.3. Constructal Design Applied to OWC–WEC

_{En}), total volumes (V

_{Tn}), and device column thicknesses (e

_{n}), where n ranges from one to five, representing the number of interconnected chambers.

_{n}) of 0.1 m. Since it is a problem in which all hydropneumatic chambers exhibit geometric similarity, it is possible to generalize their formulation; the sub-indices follow the definition already presented in this text. The dimension W in Equations (6) and (7) is kept constant and equal to 1 m because it is a two-dimensional computational model.

_{n}/L

_{n}). Using Equations (6) and (7), it is possible to determine the equations that define the lengths (L

_{n}) and the heights (H

_{n}) of the problem:

^{3}and ${V}_{{E}_{n}}$ = 65.4 m

^{3}. These values for the entrance and total volumes are presented by Gomes et al. [28], who studied the geometric recommendations of OWC devices by means of Constructal Design.

## 3. Computational Modeling

#### 3.1. The Multiphase Model

^{3}), t is the time (s), $\overrightarrow{v}$ is the flow velocity vector (m/s), p is the static pressure (Pa), $\mu $ is the viscosity (kg/m·s), $\stackrel{\u033f}{\tau}$ is the tension tensor, and $\overrightarrow{g}$ is the acceleration of gravity (m/s

^{2}).

_{air}= 1 − α

_{water}). Cells with α

_{water}= 0 are entirely filled with air (α

_{air}= 1), while, conversely, those with α

_{air}= 0 are entirely filled with water.

#### 3.2. Numerical Wave Generation and Boundary Conditions

_{1}and C

_{2}are the linear and quadratic damping coefficients, respectively, the term $\rho $ (kg/m

^{3}) refers to density, $\overrightarrow{v}\text{}$(m/s) is the velocity, z (m) is the vertical position, z

_{fs}(m) and z

_{b}(m) are the vertical positions of the top and the bottom, x (m) is the horizontal position, and x

_{s}(m) and x

_{e}(m) are the horizontal positions of the beginning and finish of the numerical beach. More details about the studies related to the linear and quadratic terms in the equations can be found in [29,42].

#### 3.3. Mesh Study

#### 3.4. Evaluation of the Numerical Model

## 4. Results and Discussion

#### 4.1. Results for Pressure and Mass Flow Rate

_{n}/L

_{n}ratio.

_{n}/L

_{n}= 0.3919, i.e., H

_{n}= 5.063 m and L

_{n}= 12.918 m.

_{n}/L

_{n}= 0.2613, with H

_{n}= 4.1335 m and L

_{n}= 15.8219 m. Furthermore, with an increase in the ratio—meaning an increase in chamber height and a decrease in chamber length—there was a reduction in the mass flow rate values. This trend is explained by the length of the duct housing the turbine of the device, which remained constant, as the mass flow rate tends to decrease even under high pressure.

#### 4.2. Results for Hydropneumatic Power

_{n}/L

_{n}. The main explanation rests on the effect of compression and decompression of the hydropneumatic chambers and the piston effect that occurs differently with the increase in the height of the chambers and the decrease in their width.

_{n}/L

_{n}= 0.2613 (H

_{n}= 5.0625 m and L

_{n}= 15.8219 m), led to an available hydropneumatic power of (P

_{hyd})

_{opt}= 30.8 kW. The lowest performance case had one chamber, and the ratio H

_{n}/L

_{n}= 0.0153 (H

_{n}= 1 m and L

_{n}= 65.4 m) conducted an available hydrodynamic power of (P

_{hyd})

_{opt}= 0.4168 kW.

_{n}/L

_{n}$\cong 171H/10\lambda $ achieves maximum energy conversion. This result is a theoretical recommendation for cases with five coupled OWC devices based on the geometry that presented the best performance (H

_{n}/L

_{n}= 0.2613).

## 5. Conclusions

_{1}/L

_{1}, H

_{2}/L

_{2}, H

_{3}/L

_{3}, H

_{4}/L

_{4}, and H

_{5}/L

_{5}. The problem restrictions were defined as the volume of the hydropneumatic chambers (V

_{E}

_{1}, V

_{E}

_{2}, V

_{E}

_{3}, V

_{E}

_{4}, and V

_{E}

_{5}), the volume of each hydropneumatic chamber added to the turbine duct volume (V

_{T}

_{1}, V

_{T}

_{2}, V

_{T}

_{3}, V

_{T}

_{4}, and V

_{T}

_{5}), and the thickness of the columns that divide the devices, which were kept constant (e

_{1}, e

_{2}, e

_{3}, e

_{4}, and e

_{5}).

_{n}/L

_{n}= 0.2613 (H

_{n}= 5.0625 m and L

_{n}= 15.8219 m).

_{n}/L

_{n}= 0.2613 (H

_{n}= 5.0625 m and L

_{n}= 15.8219 m), leading to a hydropneumatic power of (P

_{hyd})

_{opt}= 30.8 kW. The lowest performance case had one chamber, and the ratio H

_{n}/L

_{n}= 0.0153 (H

_{n}= 1 m and L

_{n}= 65.4 m) conducted a hydrodynamic power of (P

_{hyd})

_{opt}= 0.4168 kW.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Parameters | ||

V_{En} | volume of the hydropneumatic chambers | m^{3} |

V_{Tn} | total volume | m^{3} |

e_{n} | thickness of the columns | m |

H_{n} | height of the hydropneumatic chambers | m |

H_{j} | height of the turbine duct | m |

L_{n} | length of the hydropneumatic chambers | m |

l_{n} | length of the turbine duct | m |

H_{depth} | device’s submersion depth | m |

T | period of wave | s |

H | height of wave | m |

H | depth | m |

A | amplitude | m |

T | time | s |

$\overrightarrow{v}$ | flow velocity vector | m/s |

P_{e,air} | static pressure | Pa |

L_{T} | tank length | m |

H_{T} | tank height | m |

U | velocity in the x direction | m/s |

Z | velocity in the y direction | m/s |

A_{T} | area of each volume | m^{2} |

C_{1} | linear damping coefficient | - |

C_{2} | quadratic damping coefficient | - |

x, z | spatial coordinates | m |

P | pressure | Pa |

P_{hyp} | hydropneumatic power | W |

${\dot{m}}_{air}$ | mass flow rate | kg/s |

Subscript and abbreviations | ||

WEC | wave energy converter | |

OWC | oscillating water column | |

FVM | finite volume method | |

VOF | volume of fluid | |

CFD | Computational Fluid Dynamics | |

PRESTO | Pressure Staggering Option | |

PISO | Pressure Implicit Split Operator | |

RMS | root mean square |

## Appendix A

Cases | H_{n}/L_{n} | H_{n} (m) | H_{j} (m) | L_{n} (m) | l_{j} (m) |
---|---|---|---|---|---|

1 | 0.0153 | 1.0000 | 9.1698 | 65.4000 | 3.0566 |

2 | 0.0229 | 1.2247 | 9.1698 | 53.3989 | 3.0566 |

3 | 0.0344 | 1.5000 | 9.1698 | 43.6000 | 3.0566 |

4 | 0.0516 | 1.8371 | 9.1698 | 35.5993 | 3.0566 |

5 | 0.0774 | 2.2500 | 9.1698 | 29.0667 | 3.0566 |

6 | 0.1161 | 2.7557 | 9.1698 | 23.7328 | 3.0566 |

7 | 0.1742 | 3.3750 | 9.1698 | 19.3778 | 3.0566 |

8 | 0.2613 | 4.1335 | 9.1698 | 15.8219 | 3.0566 |

9 | 0.3919 | 5.0625 | 9.1698 | 12.9185 | 3.0566 |

10 | 0.5878 | 6.2003 | 9.1698 | 10.5479 | 3.0566 |

11 | 0.8817 | 7.5938 | 9.1698 | 8.6123 | 3.0566 |

**Table A2.**Geometric variation in the height of the columns that divide the devices with five coupled chambers.

Cases | H_{2}, H_{3}, H_{4}, H_{5} (m) | H_{1}/L_{1}; H_{6}/L_{5} | H_{j} (m) | L_{1}; L_{5} (m) | l_{j} (m) |
---|---|---|---|---|---|

1 | 0.0000 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

2 | 0.2419 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

3 | 0.3629 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

4 | 0.5443 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

5 | 0.8165 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

6 | 1.2247 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

7 | 1.8371 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

8 | 2.7557 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

9 | 4.1335 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

10 | 6.2003 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

11 | 9.3004 | 0.2613 | 9.1698 | 15.8219 | 3.0566 |

Cases | p (Pa) | $\dot{\mathit{m}}$ (kg/s) | P_{hid} (W) |
---|---|---|---|

1 | 2191.0142 | 16.4658 | 416.7942 |

2 | 2259.7285 | 22.2433 | 896.1300 |

3 | 2378.7092 | 28.0633 | 1341.7517 |

4 | 2558.5239 | 29.6668 | 1776.1438 |

5 | 4857.0251 | 38.0522 | 6750.6147 |

6 | 3332.7051 | 41.9347 | 4703.5118 |

7 | 6784.9656 | 61.0432 | 17,010.7276 |

8 | 8820.8559 | 82.1106 | 30,815.7217 |

9 | 9049.5421 | 68.1899 | 24,617.2238 |

10 | 6694.0390 | 55.3329 | 14,426.9606 |

11 | 6474.0304 | 49.7657 | 12,517.5821 |

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**Figure 1.**Visualization of the 2D computational model comprising the wave channel and the OWC device. (Adapted from Lima et al. [17]).

**Figure 5.**Comparison of the free water surface elevation for numerical and experimental results: (

**A**) all simulation times; (

**B**) detail of 100 s ≤ t ≤ 150 s.

**Figure 6.**Comparison among the results obtained in the present study and those presented by Liu et al. [45] for different wave periods.

**Figure 7.**Accumulated root mean square pressure in coupled device as function of height/length of OWC chambers.

**Figure 8.**Accumulated root mean square mass flow rate in each coupled device as a function of the height/length ratios of the OWC chambers.

**Figure 9.**Accumulated available root mean square hydropneumatic power in each coupled device as function of height/length of OWC chambers.

Characteristics | Values |
---|---|

Wave period (T) | 7.5 s |

Wave length (λ) | 65.4 m |

Wave height (H) | 1 m |

Depth (h) | 10 m |

Tank length (L_{T}) | 327 m |

Tank height (H_{T}) | 14 m |

Formulation | Adopted Parameters |
---|---|

Solution in Time | Transient |

First Order Implicit Formulation | |

Based on Pressure | |

Non-Iterative Advance | |

VOF Model | Explicit Formulation |

Solution Control | Pressure–Velocity Coupling Method: PISO |

Geometric Fraction Discretization Scheme: Geo-Reconstruct | |

Pressure Discretization Method: PRESTO! | |

Formulation of Momentum: First Order Implicit |

Formulation | Adopted Parameters |
---|---|

volume of fluid (VOF) | Fluids: air and water |

Volume Fraction Formulation (α) | |

finite volume method (FVM) | Mass and Momentum Conservation |

Cases | Nº of Volumes | Mean Error [%] | Simulation Time [h] |
---|---|---|---|

1 | 2,487,276 | 1.85 | 70 |

2 | 1,877,436 | 1.94 | 52 |

3 | 1,167,166 | 2.10 | 32 |

4 | 1,093,356 | 2.48 | 27 |

5 | 766,656 | 1.76 | 18 |

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**MDPI and ACS Style**

de Lima, Y.T.B.; Isoldi, L.A.; dos Santos, E.D.; Machado, B.N.; Gomes, M.d.N.; Biserni, C.; Rocha, L.A.O.
Study of the Geometry of an Oscillating Water Column Device with Five Chambers Coupled under Regular Waves through the Constructal Design Method. *Fluids* **2024**, *9*, 86.
https://doi.org/10.3390/fluids9040086

**AMA Style**

de Lima YTB, Isoldi LA, dos Santos ED, Machado BN, Gomes MdN, Biserni C, Rocha LAO.
Study of the Geometry of an Oscillating Water Column Device with Five Chambers Coupled under Regular Waves through the Constructal Design Method. *Fluids*. 2024; 9(4):86.
https://doi.org/10.3390/fluids9040086

**Chicago/Turabian Style**

de Lima, Yuri Theodoro Barbosa, Liércio André Isoldi, Elizaldo Domingues dos Santos, Bianca Neves Machado, Mateus das Neves Gomes, Cesare Biserni, and Luiz Alberto Oliveira Rocha.
2024. "Study of the Geometry of an Oscillating Water Column Device with Five Chambers Coupled under Regular Waves through the Constructal Design Method" *Fluids* 9, no. 4: 86.
https://doi.org/10.3390/fluids9040086